The concept of the sparse index tracking portfolio has attracted significant attention in the field of finance and investment management due to its advantages over full replication portfolios. In this paper, we study asset selection methods for constructing a sparse index tracking portfolio. We propose useful formulations, which are described as combinatorial optimization problems, to select the assets. By adjusting the values of the parameters, these formulations can yield various asset selection methods, including some existing methods. As a result, the proposed formulations can provide a well-balanced asset selection to create successful sparse index tracking portfolios. We also provide numerical examples to compare the tracking performance of the resulting sparse index tracking portfolios.